KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will have to demonstrate knowledge
of the fundamental object of functional analysis, both linear and
nonlinear.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING
At the end of the course the student must know how to apply the
knowledge of basic functional analysis acquired to solve problems and
exercises of medium difficulty. The exercises can also be proposed as
easy theoretical results.
JUDGMENT AUTONOMY
At the end of the course the student will be able to recognize and apply
the most basic techniques of functional analysis and also to recognize the
situations and problems in which these techniques can be used
advantageously.
COMMUNICATIVE SKILLS
At the end of the course the student will be able to express himself
appropriately on the topics of the course.
Testi in inglese
English
Spectrum of linear operators. Spectral theorem for bounded self-adjoint
operators in Hilbert spaces.
Differential calculus in Banach spaces. The implicit function theorem.
Lyapunov–Schmidt reduction and bifurcation.
Theory of topological degree and fix point theorems: Brouwer, Schauder.
LEARNING CAPACITY
At the end of the course the student should be able to consult the
standard texts of functional analysis, both linear and nonlinear

Basic functional spaces: continuous functions and Lebesgue space; weak
topologies.
Fundamental theorem of functional analysis (Hanh-Banach, Baire, open
mapping, closed graph).

Spectrum of linear operators. Spectral theorem for bounded self-adjoint
operators in Hilbert spaces.

Differential calculus in Banach spaces. The implicit function theorem.

Lyapunov–Schmidt reduction and bifurcation.

Theory of topological degree and fix point theorems: Brouwer, Schauder.

A. Ambrosetti, G. Prodi: A primer of nonlinear analysis, Cambridge,
Cambridge University Press 1993.
H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential
Equations, Springer-Verlag New York, 2011.
K. Deimling: Nonlinear Function Analysis, Springer-Verlag Berlin
Heidelberg, 1984.
M. Reed, B. Simon: Methods of Modern Mathematical Physics I -
Functional analysis, Elsevier, 1972.

Part 1) Linear Analysis.
Linear bounded operators, adjoints. Compact operators and Fredholm’s
alternative Theorem. The spectrum of an operator and its topological
properties. Spectrum of self-adjoint operators. Spectral theory for
compact and self-adjoint operators. Min-max Fisher-Courant Theorem.
Spectral theorem for self-adjoint bounded operators: continuous and
Borellian functional calculus.
Applications to Sturm-Liouville problems.

Part 2) Nonlinear Analysis.
Differential Calculus in Banach spaces. Gateaux and Frechet
differentiable functions, examples. Mean value Theorem. Higher order
derivatives. Differentiability of Nemiski operators. Inverse function
Theorem and Implicit function Theorem. Applications to non-linear
problems and to ODE in Banach spaces. Lagrange multiplier Theorem
and applications. Bifurcation Theorem, necessary condition of bifurcation,
Lyapunov-Schmidt reduction. Bifurcation of the simple eigenvalue.
Topological degree theory, Brouwer and Leray-Schauder degree and fixed
points

Lectures and problem sessions. During the course some exercises will be discussed in class.

The exam program coincides with the arguments of the lectures. The exam will be written + oral consisting in verifying the comprehension of the contents (definitions and proofs) and the ability in explaining the subject and to correctly apply the theory.